Limit Cycle Behaviour of the Bump-on-Tail Instability

The nonlinear dynamics of the bump‐on‐tail instability is considered. The eigenmodes have discrete k because of finite periodic boundary conditions. Increasing a critical parameter (the number density) above its neutral stable value by a small fractional amount Δ2, one mode becomes unstable. The nonlinear dynamics of the unstable mode is determined by means of the multiple time scale method.

Usually, limit cycle behavior is found. A short comparison with quasi‐linear theory is given, and the results are compared with experiment.